Short version: We don't know the answer to either question in any definitive sense. Every warship we have details of has vague gaps in their battle history. For example we might know that they participated in one of the times Terra was liberated, yet not know how many "kills" they racked up. We rarely have specific details (particularly on who scored what kills) except for a few very specific battles that are well-detained in the novel fiction or sourcebook material. These are usually smaller battles where we get something like a play-by-play of the entire battle.
Longer version:
The answer to the latter question is probably one of the Clan warships with history dating all the way back to the SLDF. The trick is they've mostly been renamed and their battle histories aren't fully known. But there's no doubt in my mind that some of those ships that were part of Kerensky's campaign to liberate Terra racked up some kills, and then continued to get a few more in the centuries since, particularly through any combat they saw in the Jihad/Wars of Reaving era.
The answer to the first question is probably some Terran Hegemony vessel (possibly turned SLDF) from back when warships were plentiful. Think some "Age of War" vessel. In those days there were a lot of warships flying around, the Terran Hegemony, all the Great Houses, everyone had some, and accounts of their various combat exploits only fill the margins of some books here or there. There is evidence that warship combat was a lot more common and routine in those early days and that naval power (and actively using it) was a priority for the Terran Hegemony. Not just titanic clashes, but more common border fighting, including significant naval battles and the like that the books have not covered in complete detail.
Going farther out on a speculative limb, the answer may be some Aegis-class vessel. Think about how long that class has been in service. There are probably some hulls of that class that have been cycled through many times. Fought in the Terran Hegemony navy, got refitted and fought with the SLDF in the Reunification War, got refitted again and fight in the Amaris Coup. Just by sheer time in service and the number of conflicts they've survived through, they've had the most opportunity at least, to rack up the most kills.
Although I can't stress enough, there was a lot of common warship fighting in the Terran Hegemony days. It's very possible the top-scoring winner lived its entire service life.. .from commissioning to death or scrapped, before the Star League even existed. Part of me thinks it would be kinda strange and yet appropriate if it was a class of vessel that hasn't been around in more modern Battletech time, like a Monsoon-class Battleship. According to TRO 3057 Revised, at the time several of the Monsoon-class named vessels were legendary and had legendary names among the navy and the SLDF Admirals, full of nostalgia for these ships, tried to keep them in service for as long as possible. It also describes them as once having terrorized enemy fleets in the Age of War era, which makes it feel like they did do a lot of fighting.
Statistically speaking, it's highly unlikely that the top-scoring winner of all time exists in the ilClan era. The vast majority of warships ever built in the Battletech universe no longer exist. To me it's a slim bet that a top-scoring warship of all time still survives.
